存在噪声和离群点时特征匹配图的最优检测
Optimal detection of the feature matching map in presence of noise and outliers
摘要：
我们考虑的问题，找到匹配映射之间的两组$d$维向量从噪声观测，其中第二组包含离群值。匹配映射则是注入，只有当第二组向量被很好地分离时，才能一致地估计该注入。主要结果表明，在高维环境中，未知注入的检测区域可以由内点-内点距离至少为$d^{1/4}$阶且内点-外点距离至少为$d^{1/2}$阶的向量集来表征。这些速率是使用最小化匹配点对之间的距离的平方和的估计匹配来实现的。我们还证明了建立这些利率的最优性的下界。最后，我们报告的合成和现实世界的数据，说明我们的理论结果，并提供进一步的洞察力在这项工作中研究的估计的属性的数值实验结果。
英文摘要：
We consider the problem of finding the matching map between two sets of $d$ dimensional vectors from noisy observations, where the second set contains outliers. The matching map is then an injection, which can be consistently estimated only if the vectors of the second set are well separated. The main result shows that, in the high-dimensional setting, a detection region of unknown injection can be characterized by the sets of vectors for which the inlier-inlier distance is of order at least $d^{1/4}$ and the inlier-outlier distance is of order at least $d^{1/2}$. These rates are achieved using the estimated matching minimizing the sum of logarithms of distances between matched pairs of points. We also prove lower bounds establishing optimality of these rates. Finally, we report results of numerical experiments on both synthetic and real world data that illustrate our theoretical results and provide further insight into the properties of the estimators studied in this work.
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